Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups

Rump, Wolfgang

doi:10.4153/s0008439521000230

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…Then Remark. Theorem 1 is proved, but not correctly stated, in [27] where the condition e 2 X 2 T .A/ is replaced by the stronger assumption that e generates A. By [25, Theorem 3], both conditions are equivalent if the multipermutation level of X is finite.…”

Section: Uniconnected Cycle Setsmentioning

confidence: 99%

The classification of nondegenerate uniconnected cycle sets

Rump

2023

Pacific J. Math.

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It is known that the set-theoretic solutions to the Yang-Baxter equation studied by Etingof et al. (1999) are equivalent to a class of sets with a binary operation, called nondegenerate cycle sets. There is a covering theory for cycle sets which associates a universal covering to any indecomposable cycle set. The cycle sets arising as universal covers are said to be uniconnected. In this paper, the category of nondegenerate uniconnected cycle sets is determined, and it is proved that up to isomorphism, a nondegenerate uniconnected cycle set is given by a brace A with a transitive cycle base (an adjoint orbit which generates the additive group of A). The theorem is applied to braces with cyclic additive or adjoint group, where a more explicit classification is obtained..x y/ .x z/ D .y x/ .y z/ MSC2020: 05E18, 16T25, 81R50.

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Section: Uniconnected Cycle Setsmentioning

confidence: 99%

The classification of nondegenerate uniconnected cycle sets

Rump

2023

Pacific J. Math.

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Section: Uniconnected Cycle Setsmentioning

confidence: 99%

Untitled

2023

Pacific J. Math.

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“…Many authors focused on studying and classifying such solutions with a special emphasis on those which are also involutive and non-degenerate, i.e. solutions (X, r) such that r 2 = id, and if we write r(x, y) = (σ x (y), τ y (x)) then the maps σ x and τ y are bijective, for all x, y ∈ X; see [6,7,8,9,18,21,25,33,34,36,37,39]. On the other hand, almost nothing is known about indecomposable solutions which are non-degenerate and non-involutive (see [13,26,32]).…”

Section: Introductionmentioning

confidence: 99%

On derived-indecomposable solutions of the Yang--Baxter equation

Colazzo¹,

Ferrara²,

Trombetti³

2022

Preprint

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If (X, r) is a finite non-degenerate set-theoretic solution of the Yang-Baxter equation, the additive group of the structure skew brace G(X, r) is an F C-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an F C-group itself. If one additionally assumes that the derived solution of (X, r) is indecomposable, then for every element b of G(X, r) there are finitely many elements of the form b * c and c * b, with c ∈ G(X, r). This naturally leads to the study of a brace-theoretic analogue of the class of F C-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.

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Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups

Cited by 3 publications

References 31 publications

The classification of nondegenerate uniconnected cycle sets

The classification of nondegenerate uniconnected cycle sets

Untitled

On derived-indecomposable solutions of the Yang--Baxter equation

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